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Rationality and Operators

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Rationality and Operators

Part of the book series: SpringerBriefs in Economics ((BRIEFSDBJRS))

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Abstract

This chapter examines the operational implications of rationality concepts. First, we introduce various types of rationality concepts and examine their relationship. We emphasize how they can be represented by operators over binary relations. Second, we introduce and examine the concept of a choice function, which is a consequence of operations over some binary relation. Third, we discuss extensions of a binary relation.

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Notes

  1. 1.

    See, for example, Luce (1956), Sen (1969 ) , and Fishburn (1973, 1975, 1997 ) .

  2. 2.

    Bossert and Suzumura (2010 ) also divide properties of binary relations into two classes: the first is the class of richness properties, and the second is the class of coherence properties. Basically, the class of simple properties corresponds to richness properties, and the class of composite properties corresponds to coherence properties . However, the correspondence is an approximate one. First, symmetry, asymmetry, and antisymmetry are simple, but it is difficult to regard them as richness properties. Second, semi-transitivity and the interval-order property are composite properties, but they also have some aspects in common with richness properties.

  3. 3.

    See Suzumura (1983) and Bossert and Suzumura (2010).

  4. 4.

    Armstrong (1939 ) provides an early criticism of transitive indifference. A well-known criticism is the coffee-sugar example of Luce (1956 ) . See Chap. 1.

  5. 5.

    See Luce (1956) and Blau (1979 ) . However, their formulation is as follows: \(\Big ( P(R) \circ P(R) \circ I(R) \Big ) \subseteq P(R)\).

  6. 6.

    The money-pump argument is attributed to Raiffa (1968).

  7. 7.

    For a comprehensive argument on choice functions, See Sen (1970), Suzumura (1983 ) , and Moulin (1985 ) .

  8. 8.

    Bradley (2013 ) examines another construction of a choice function. Intuitively, his concept is stronger than acyclicity (under reflexivity). As shown by Bradley (2013), it is strongly related to Suzumura consistency.

  9. 9.

    See Sen (1970).

  10. 10.

    See Plott (1973 ) and Suzumura (1983). They clarify this point by employing the revealed preference approach.

  11. 11.

    In strong path independence, only triplets are considered to be path independent. Bandyopadhyay (1988) considers more general choice procedures for each menu \(A \subseteq X\).

  12. 12.

    The analysis of this section is based on Cato (2012a), which provides alternative proofs of extension theorems. The proofs of Propositions 3.8, 3.10, and 3.12 are modeled after Cato (2012a).

  13. 13.

    Duggan (1999 ) shows (i) of Lemma 3.23. See also Cato (2012a).

  14. 14.

    Hansson (1968) provides the formal proof.

  15. 15.

    Moore (1982 ) provides an introduction to the axiom of choice and related subjects.

  16. 16.

    Aliprantis and Border (2006 ) explain the implications and significance of Zorn’s Lemma.

  17. 17.

    See Chap. 5 and Cato (2012b).

  18. 18.

    Inada (1954 ) provides an alternative proof of Arrow’s extension theorem.

  19. 19.

    Arrow (1951) assumes that \(R|_A \subseteq \Delta \). That is, his assumptions are stated as follows: Let R be a quasi ordering on X, let \(A \subseteq X\) be such that \(R|_A \subseteq \Delta \), and let \(R'\) be a ordering on A. Note that \(R'\) is an extension of \(R|_A\) since \(R'\) is complete on A. Then, Proposition 3.10 is a generalization of Arrow’s result. Our proof is modeled after Cato (2012a), which presents an alternative proof of Arrow’s result.

  20. 20.

    A binary relation R is monotonic if, for all \(x,y \in X\),

    $$ x \ge y \Rightarrow (x,y) \in R. $$

    .

  21. 21.

    This result is a generalization of Theorem 2.9 of Bossert and Suzumura (2010) and the theorem of Suzumura (2004).

  22. 22.

    Cato (2012a) assumes that \(R|_A \subseteq \Delta \). The proof is modeled after Cato (2012a).

  23. 23.

    Sen (1995) also discusses the significance of preferences in this class.

  24. 24.

    See Debreu (1954).

  25. 25.

    Scott and Suppes (1958 ) provides the representation theorem.

  26. 26.

    See Fishburn (1973 ) , Ok (2002 ) , and Andrikopoulos (2007 ) .

  27. 27.

    Suzumura (1983) and Bossert and Suzumura (2010) cover the topics of revealed preference theory.

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Authors and Affiliations

  1. Institute of Social Science, The University of Tokyo, Hongō, Bunkyō, Tokyo, Japan

    Susumu Cato

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Cato, S. (2016). Rationality and Operators. In: Rationality and Operators. SpringerBriefs in Economics(). Springer, Singapore. https://doi.org/10.1007/978-981-10-1896-1_3

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